The song you are talking about was the theme tune of the TV series Mission: Impossible (1966–1973). It returned to television for two seasons from 1988 to 1990.
Original theme From YouTube:
The Mission Impossible movie is based on the TV series.
In 1996, the theme was remade by U2 members Adam Clayton and Larry Mullen, Jr....
According to the director, the actor in the photograph is the same person that played Ray Finkle for the rest of the movie
merely costumed up to look more like a typical 70's football player.
To quote from the article:
I can't find any backup, but I read somewhere way-back-when that they needed a mascot who could (possibly) kick a field goal, to ridicule the actual kicker who missed his attempt. The dolphin is the only mascot capable of doing something like that.